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Physics · Glossary

What is Huygens–Fresnel principle?

Definition 22.1 University Physics — Year 2 · Chapter 22 — Fraunhofer Diffraction and Spatial Filtering

Every point of a wavefront (or of an aperture lit by a wave) acts as a secondary source emitting a spherical wavelet, in phase with the wave there; the wave beyond is the coherent sum of the wavelets. For an aperture Σ\Sigma lit by a plane wave of amplitude aa, the amplitude reaching a far point is s‾(M)=K∬Σe−iφ(P,M) ⁣dS\underline s(M) = K\iint_\Sigma\eu^{-\iu\varphi(P, M)}\dd S, with φ(P,M)\varphi(P, M) the phase of the path from the aperture point PP to MM (KK a constant). Fraunhofer (far-field) diffraction is the case where MM is at infinity — or in the focal plane of a lens — so that the rays from all points of the aperture toward MM are parallel, in the direction u→\vect u, and

φ(P,M)=φ0−2πλ OP→⋅u→,s‾(u→)∝∬Σei(2π/λ)OP→⋅u→ ⁣dS.\varphi(P, M) = \varphi_0 - \frac{2\pi}\lambda\,\vect{OP}\cdot\vect u , \qquad \underline s(\vect u) \propto \iint_\Sigma\eu^{\iu(2\pi/\lambda)\vect{OP}\cdot\vect u}\dd S .

The pattern is observed at infinity, or at the focus F′F' of a lens, where the direction u→\vect u of small angles (α,β)(\alpha, \beta) maps to the point (fα,fβ)(f\alpha, f\beta).

Examples

Example 22.5 (Eye, telescope, microscope)

The eye (D=3 mmD = 3\,\mathrm{mm}, λ=550 nm\lambda = 550\,\mathrm{nm}): 1.22λ/D=2.2×10−4 rad1.22\lambda/D = 2.2 \times 10^{-4}\,\mathrm{rad}, about 45′′45'' — a 1 mm1\,\mathrm{mm} detail at 4 m4\,\mathrm{m}, which is indeed about the acuity of a good eye: evolution matched the retina’s cells to the diffraction limit. A 2.4 m2.4\,\mathrm{m} telescope: 0.06′′0.06'', a thousand times better, if the atmosphere lets it (it does not, from the ground: 1′′1'' of "seeing", whence space telescopes and adaptive optics). A radio dish of 100 m100\,\mathrm{m} at λ=21 cm\lambda = 21\,\mathrm{cm}: 9′9', worse than the eye; interferometers across continents restore the resolution. A microscope objective of aperture angle α\alpha resolves 0.61λ/nsin⁡α0.61\lambda/n\sin\alpha — about 0.2 µm0.2\,\text{µ}\mathrm{m} in the visible, whatever the magnification: the limit that drove microscopy to electrons and to X-rays.

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